91Ó°ÊÓ

To solve the indefinite integral \( \int \frac{x^{n-1}}{1+x^{2n}} \ dx \), the substitution method is a powerful tool. This method transforms the integral into a simpler form that is easier to evaluate. The main idea behind substitution is to set a part of the integral as a new variable, which simplifies the function being integrated. In this exercise, we perform the substitution \( u = x^n \). When you choose \( u = x^n \), the differential becomes \( du = n \cdot x^{n-1} \, dx \), or equivalently, \( x^{n-1} \, dx = \frac{du}{n} \). Substituting back, the original integral changes into \( \int \frac{1}{1+u^2} \cdot \frac{du}{n} \), which is significantly simpler to evaluate. The substitution method works well when the integral contains a function and its derivative. Here, \( x^{n-1} \) is the derivative part that complements \( x^n \), allowing us to easily switch to the \( u \)-variable. Remember, the goal of substitution is to change the integral into a standard form or make it much simpler than the original.

Once the substitution is made and the integral is transformed into \( \frac{1}{n} \int \frac{du}{1+u^2} \), it's time to recognize the inverse tangent function. This form of the integral corresponds directly to a well-known result: the inverse tangent function. The integral \( \int \frac{du}{1+u^2} \) results in \( \arctan(u) + C \), where \( C \) is the constant of integration. The reason \( \frac{1}{1+u^2} \) suggests the inverse tangent is that its derivative, \( \frac{d}{du} \arctan(u) \), equals \( \frac{1}{1+u^2} \). In this context, "arctan" is an abbreviation for "arc tangent," a specific type of inverse trigonometric function. It essentially gives the angle whose tangent is the original value. This function is critical in this exercise because plugging \( u = x^n \) back into \( \arctan(u) \) completes the integration process and forms part of the final answer.

The constant of integration, denoted as \( C \), is an integral part of evaluating indefinite integrals. Whenever you find an indefinite integral, such as \( \int \frac{du}{1+u^2} = \arctan(u) + C \), the constant \( C \) must be included in your final answer. The reason for \( C \) is tied to the nature of indefinite integrals. They represent a family of functions, not just a single function. Any constant added to a function does not affect its derivative. Hence, adding \( C \) accounts for all constant shifts that yield the same differentiation result, as integrals reverse the process of differentiation. For this reason, omit \( C \) at your peril! It is important to reinforce this idea in the solution as not including \( C \) renders the solution incomplete. By convention, mathematicians insert \( C \) to preserve all possible original function forms that would differentiate to the given function in the integral.